Extending Maps in Hilbert Manifolds
Volume 60 / 2012
Abstract
Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if $X$ is a completely metrizable space of topological weight not greater than $\alpha \geq \aleph_0$, $A$ is a closed set in $X$ and $f\colon X \to M$ is a map into a manifold $M$ modelled on a Hilbert space of dimension $\alpha$ such that $f(X \setminus A) \cap \overline{f(\partial A)} = \emptyset$, then for every open cover $\mathcal U$ of $M$ there is a map $g\colon X \to M$ which is $\mathcal U$-close to $f$ (on $X$), coincides with $f$ on $A$ and is an embedding of $X \setminus A$ into $M$. If, in addition, $X \setminus A$ is a connected manifold modelled on the same Hilbert space as $M$, and $\overline{f(\partial A)}$ is a $Z$-set in $M$, then the above map $g$ may be chosen so that $g|_{X \setminus A}$ be an open embedding.
